Large Eddy Simulation of Titanium Dioxide Nanoparticle Formation and Growth in Turbulent Jets

Abstract

Large eddy simulations (LES) of titanium dioxide nanoparticles in three dimensional turbulent reacting planar jets are performed. The spatio-temporal evolution of the particle field is obtained by utilizing a nodal representation of the general dynamic equation. Gradient-diffusion, Smagorinsky-type subgrid-scale closures are employed to account for the unresolved stresses, fluid-scalar fluxes, and fluid-particle fluxes. The effect of the unresolved fluctuations on coagulation are neglected. Simulations are performed at two different precursor concentration levels. Comparison between results obtained via direct numerical simulation (DNS) and LES is performed to assess the performance of the closures. The LES performs fairly well in predicting the particle concentration as a function of size as well as the mean diameter. Additionally the polydispersity of the LES particle field is greater than that of the DNS. The results also suggest that at as the precursor concentration increases, neglect of the unresolved particle-particle interactions may act to increase the nanoparticle growth-rate.

1. Introduction

Nanoscale particles play an integral role in a wide variety of physical/chemical phenomena and processes. There are several technologies which can be employed in the manufacture of nanoscale materials (films, particles, etc.) (Ulrich 1971); Zachariah et al. 1995; Kuo and Rivera (2007); Tricoli and Pratsinis (2009). Vapor-phase methodologies are favored because of chemical purity and cost considerations (Pratsinis 1998). The formation of very fine particles from vapor encompasses a large number of physical/chemical phenomena. The exact mechanisms affecting particle properties is a complex interaction of time-temperature history along with the methods of mixing (Vemury and Pratsinis 1995; Vemury et al. 1997). Nanoparticle synthesis under turbulent flow conditions is considered to be a efficient means for the high-rate fabrication of nano-structured materials—particles and films (Pratsinis 1989; Wooldridge 1998).

A review of the computational fluid dynamics literature identifies three general approaches for analytical description of turbulent flows (Knight and Sakell 1993; Libby and Williams 1994; Pope 1990): (1) Reynolds averaged simulations (RANS), (2) DNS, and (3) large eddy simulation (LES). The first approach has been the most popular for prediction of engineering flows due to its relatively low computational cost (Libby and Williams 1980; Pope 1985; Kollmann 1980; Jones 1994). The second approach has received significant attention within the past 15 years but its use is limited to basic research problems due to its extensive computational requirements (Oran and Boris 1981, 1987; Jou and Riley 1989; Borghi and Murthy 1989; Givi 1989). The third approach is regarded as lying somewhere between the first two (Love 1979; Voke and Collins 1983; Rogallo and Moin 1984; Schumann and Friedrich 1986; Ferziger 1987). LES has the advantage of DNS in that it captures (or tries to capture) the unsteady evolution of the large scale flow features, and it has the advantage of RANS in that it allows for inclusion of realistic flow and chemistry parameters. Of course, the closure problem as encountered in RANS is also present in LES. However, since only the effects of the small scales are modeled in LES the closure uncertainties at such scales are less damaging than those in RANS in which modeling is applied at all scales. While there has been significant progress in development and implementation of LES in turbulent, single-phase flows, comparatively little has been done in its utilization in turbulent aerosols (Lumley 1990; Moin 1991; Jaberi et al. 1999; Gicquel et al. 2002; Givi 2006; Lambert et al. 2011). There have been recent efforts towards the simulation of nanoparticle dynamics in turbulent flows. These include both DNS as well as other modeling approaches (Settumba and Garrick 2003; Miller and Garrick 2004; Wang and Garrick 2005; Garrick et al. 2006; Settumba and Garrick 2007; Das and Garrick 2010). These have been useful in illustrating, for example, the dynamics of nucleation, condensation, and coagulation under a variety of flow conditions. Unfortunately the compute times for DNS range into the hundreds of thousands of CPU-hours, rendering it unfeasible for the simulation of “real world” flows. While RANS has received some attention, LES has not been widely developed or utilized for nano-scale multiphase flows (Marchisio and Fox 2005; Schwarzer et al. 2006; Rigopoulos 2007; Zucca et al. 2007; Kartushinsky et al. 2010). LES and RANS offer more affordable compute-times but true assessments of their predicability are limited. This is especially important as one can adjust the performance of different approaches via “model constants.” Approaches which are computationally affordable and exhibit a high degree of fidelity to the physics of nanoparticle dynamics in turbulent flows are needed.

In this study, a nodal approach is employed to approximate the aerosol general dynamic equation and obtain the spatio-temporal evolution of the particle field. That is, the spatially-filtered nodal equations are solved in conjunction with the spatially-filtered Navier-Stokes equations. The results of direct numerical simulation is utilized in assessing the performance of the spatially-filtered governing equations for the nodal approximation to the GDE. Particularly, we attempt to perform LES titanium tetrachloride hydrolysis, and the subsequent growth of titanium dioxide nanoparticles in incompressible, three-dimensional planar jets. Our goals are two-fold: (1) to show a clear and mathematically consistent approach to utilizing LES in conjunction with nodal/sectional approaches and (2) to predict the behavior of the turbulent fluid-particle system undergoing formation and growth processes. The LES is performed using zero-equation, Smagorinsky-type closures. These models are used to represent the effect of the sub-grid motions in the fluid field and in advecting the particle concentrations. The effect of the subgrid-scale particle-particle interactions are neglected however. Instantaneous and statistical analysis are performed to provide both a qualitative and a quantitative assessment of the LES.

2. Formulation

2.1. Fluid Transport

The flows under consideration are incompressible shear flows containing nanoscale particles. The primary transport variables for the fluid field are the velocity vector and the fluid pressure , where denotes the spatial coordinates. These variables are governed by the conservation of mass and momentum equations

where ρ is the fluid density, and ν is the kinematic viscosity.

2.2. Chemical Transport

The mass fraction of N_s species i, Y_i , (i=1, 2, …, N_s ), is obtained by solving the species transport equation, given by

where is the diffusion coefficient of species i and represents the rate of creation or consumption of species i due to chemical reactions.

2.3. Particle Transport

Particle transport is governed by the aerosol general dynamic equation (GDE). Generally the GDE describes particle dynamics under the influence of various physical/chemical phenomena: convection, diffusion, surface growth, coagulation, nucleation, and other internal/external forces (Friedlander 2000; Gelbard 1979). The GDE is written in discrete form as a population balance on each cluster, or particle size. However, solving the GDE directly is computationally unfeasible, except for a small range of discrete particle sizes. To overcome this difficulty, we use a nodal approach to approximate the GDE (Gelbard and Seinfeld 1980; Gelbard et al. 1980; Hounslow et al. 1988; Lehtinen and Zachariah 2002; Biswas et al. 1997; Modem et al. 2002; Modem and Garrick 2003). This approach discretizes in particle volume space and divides the entire particle size distribution into N_B “bins.” The GDE is therefore solved as a set of N_B transport equations (Xiong and Pratsinis 1993; Wang and Garrick 2005). The general transport equation for the concentration of particles of volume υ _k in bin k, Q_k , is given by

where is the particle diffusivity given by

where k_b is the Boltzmann constant, C_c is the Cunningham correction factor, and d_p is the particle diameter. (Reist 1993; Friedlander 2000; Matsoukas and Friedlander 1991; Fuchs 1964; Mackowski 2006). This diameter is based on the work of Mackowski (2006) and is given by where d_o is the diameter of the primary particle. The source term, , represents the rate of particle formation and growth, due to the combined effects of nucleation, condensation and coagulation, and is given by

where J represents the formation of monomers (in bin k = 1) (Frenklach and Harris 1987; Landgrebe and Pratsinis 1990). The Brownian collision frequency function, β _ij , is given by

where a=(3υ _o /4π)^1/6(6k_bT/ρ _p )^1/2 and . Here, υ _o is the primary particle volume, υ _i is the volume of an agglomerate in bin i, ρ _p is the particle density, n_i and n_j are the number of primary particles in bins i and j, respectively, and D_f is the fractal dimension, which is introduced to describe the shape of the agglomerate (Mulholland et al. 1988; Mackowski 2006). The nodal method is discretized in size space such that the volume of particles in two successive bins is doubled, i.e., υ _k =2×υ _k−1 (Gelbard et al. 1980). Because of the logarithmic spacing of particle sizes, collisions of two particles typically result in a size that lies between bins. The operator χ _ijk , given by

“splits” such particles into neighboring bins in a way that the particle mass and number are conserved.

2.4. Large-Eddy Simulation

The practical goal of LES is to facilitate the affordable computation of turbulent flows by explicitly solving for the large scales of motion while modeling the small scales (Smagorinsky 1963). This involves the spatial filtering operation (Aldama 1990; Germano 1992),

where x defines the spatial coordinates, h denotes the spatial filter function of width Δ _H , and ⟨f(x, t)⟩ _L represents the filtered value of the transport variable f(x, t). The fluctuations are obtained via subtraction, e.g., f′=f−⟨f⟩ _L . A thorough description for obtaining the governing equations may be found elsewhere and the governing equations for incompressible flows are simply stated below for brevity (Aldama 1990; Sagaut 2001; Colucci et al. 1998; Pope 2000). The filtered conservation of mass and momentum equations are given by

and

where ⟨u_i⟩_L, is the filtered or large-scale velocity in the i direction, ⟨p⟩_L is the filtered pressure, and τ _ij is the familiar SGS stress which represents the effects of the unresolved, small-scale velocity fluctuations given by

For each species we solve the filtered transport equation given by

where ⟨Y_i ⟩ _L is the filtered species mass fraction of the ith species, is the SGS fluid-species flux which represents the unresolved, or small-scale, interactions between the fluid and species fields

A similar process is followed to obtain the transport equation for the spatially-filtered, or large-scale, particle concentration,

where ⟨Q_k⟩_L is the filtered particle number concentration in bin k and is the filtered coagulation source term, and is the SGS fluid-particle flux given by

This term represents the effect of turbulence or the unresolved components on the fluid velocity and particle concentration, respectively (Rogallo and Moin 1984). Note that in the context of Reynolds-averaged Navier-Stokes modeling, this flux is However, the difference between the spatial-filtering and ensemble-averaging (denoted with the overbar) operations produces extra terms, i.e., (Aldama 1990; Ferziger 1977, 1981; Sagaut 2001). In order to close the set of filtered equations, models are needed for the SGS stress, the SGS fluid-species flux, the SGS fluid-particle flux, and the filtered coagulation source term

2.4.1. Subgrid-scale Stress Closure

The eddy or SGS viscosity concept is used to close the SGS stress τ _ij . The SGS stress is assumed to be proportional to the filtered rate of strain and is given by

where ⟨S_ij ⟩ _L is the resolved strain rate tensor given by

and ν _t is the SGS viscosity (Smagorinsky 1963). In this work, we use the modified kinetic energy viscosity (MKEV) model to determine the SGS viscosity ν _t . This model uses an additional test-filter, denoted to define a kinetic energy which is then used to specify the SGS or eddy-viscosity (Colucci et al. 1998). The SGS viscosity is given by

where and is a reference velocity in the x_i direction. The subscript L′ denotes the filter at the secondary level which has a characteristic size (denoted by ) larger than that of grid-level filter. This model is essentially a modified version of that proposed by Bardina et al. (Bardina et al. 1983), in which the grid and secondary filters are of equal sizes. The MKEV model works reasonably well in both transitional and turbulent regimes and has been shown to be effective in both two and three-dimensional flows (Colucci et al. 1998; Garrick et al. 1999). Approaches such as the dynamic approach of Germano can easily be substituted (Germano 1992; Garrick 1995; Germano 1996). The MKEV approach is simpler in that it does not require dynamic filtering and is therefore less compute-intensive (Germano et al. 1991).

The filtered coagulation source term is closed by assuming that the growth terms depend solely on the filtered particle number concentrations, i.e., . To be precise, we are assuming that the SGS component of coagulation, , given by

is negligible, i.e., . This assumption is made solely on the basis of simplicity and preliminary work suggests that this term may not be negligible (Weier 2002). However, accounting for the small-scale fluctuations is not straightforward, and in this work we take the filtered coagulation source term to be

2.4.2. Species Field Closure

The gradient-diffusion approximation is also used for closure of the SGS fluid-species fluxes (Eidson 1985):

where Γ _t =ν _t /Sc_t is the SGS species diffusivity and Sc_t is the SGS Schmidt number, assumed to be constant. Closure requires the specification of Sc_t and, for scalars, its value is typically taken to be between 0.25 and unity (Pope 2000; Colucci et al. 1998; Givi 2006).

2.4.3. Particle Field Closure

The SGS fluid-particle flux is closed in a manner similar to the SGS fluid-species flux:

where is the SGS particle diffusivity and is the SGS particle Schmidt number.

3. Results

3.1. Flow Configuration

The flows under consideration consist of a three-dimensional, planar jet of diameter D, with a velocity of U_o ?></tex−math></inline−formula>issuingintoaco−flowingstreamwithvelocity<inlinematheqn><equation><texstructure><?TeXU _∞. The velocity ratio is U _∞/U_o =0.2. A schematic of the flow configuration is shown in Figure 1. The Reynolds number, Re_D , based on the diameter, D, the velocity of the jet, U_o , and the viscosity, ν, is Re_D =U_oD/ν=3000. Random perturbations with maximum intensity of 5% have been added to the cross-stream velocity, v, in the shear layers at the inlet to accelerate the development of large-scale structures. The jet is composed of titanium tetrachloride (TiCl ₄) diluted in nitrogen (N ₂), and the co-flowing stream is composed of water vapor (H ₂ O) diluted in N ₂. Two cases are considered. In case 1, the jet is 0.5% TiCl ₄ and 99.5% N ₂ by mass, whereas in case 2, it is 1.0% TiCl ₄ and 99% N ₂ by mass. The precursor species TiCl ₄ and H ₂ O are present in stoichiometric amounts. The temperature is constant at T=300K. The Prandtl number, Pr, is assumed to be that of the carrier gas N ₂, i.e., Pr= 0.71.

FIG. 1 Flow configuration.

3.2. Physical Assumptions

The reaction between titanium tetrachloride and water vapor is considered to be an irreversible, one-step, isothermal, chemical reaction occurring at a temperature of 300K and atmospheric pressure and the products of this reaction are titanium dioxide (TiO ₂) and hydrochloric acid (HCl)

The diffusion coefficients of species i, , are obtained via a binary diffusion model, assuming the second species is N ₂ (Reid et al. 1977). The values of the Schmidt numbers of species i, Sc_i , are calculated using , and the values are = 1.57 and Sc_HCl = 0.93. The reaction rates are obtained via kinetic collision theory and yield very high convective Damköhler numbers—of the order of 10⁵. As a result, the molar concentrations of the species are calculated by utilitizing the infinite-rate chemistry assumption (Roquemore et al. 1986). That is, it is assumed that the reactant in lesser concentration is consumed immediately when the two reactants are brought into contact. Thus the source terms for mass fraction transport equations, Equation (3), are given by

where MW_i is the molecular weight of species i, and the constant C_i is −1, −2, and 4 for TiCl ₄, H ₂ O, and HCl, respectively. For TiO ₂, the species source term takes into account the effects of both chemical reaction and condensation of vapor on the TiO ₂ particles and is given by

where represents the rate of formation of TiO ₂ vapor due to chemical reaction and represents the rate of conversion of vapor into particles by condensation,

Here, P_N =2.89e26 represents the number of particles in 1 Kmol of TiO ₂ vapor. We assume that as TiO ₂ is produced, it appears in the form of 0.5nm diameter monomers, populating bin k=1. The fractal dimension, D_f , has been studied extensively in previous studies and it's value ranges from D_f =1 to D_f =3, depending on details of the agglomerate formation process (Rogak and Flagan 1992; Friedlander 2000). In this study, we use D_f =1.8 for particles in bin 3 and larger, D_f =3 for collisions between monomers, and D_f =2 for collisions between monomers and dimers. We assume that nucleation and condensation are instantaneous processes. This implies that the condensable species in the simulations is TiO ₂ “vapor.” The nucleation rate, J, in Equation (6) is given by

In the LES all source terms, Equations (26)–(29), are obtained by substituting the filtered species, ⟨Y_i ⟩ _L , and particle concentrations, ⟨Q_i ⟩ _L .

3.3. Numerical Specifications

Computations are performed on a domain size of 20D×15D×4D in the x, y, and z directions. The grid resolution is 512×384×128 for the DNS and 160×120×32 for the LES and was arrived at based on grid-independence studies; this is typical for simulations of similar flow-fields (DesJardin and Frankel 1998; Gicquel et al. 2002; Das and Garrick 2010). The profiles of the streamwise u-velocity and the species mass fractions, and , are specified at the inlet boundary and zero-derivative conditions are implemented at the exit. In the cross-stream direction, zero-derivative boundary conditions are used, whereas periodic boundary conditions are used in the z-direction. Twenty two bins (N_B =22) are used to discretize the particle field. This allows the solution of particles covering a range of 6 orders of magnitude in volume, with the volume of the smallest particle being υ₁=0.0654nm ³. The governing transport equations are solved using a second order accurate in time and fourth order accurate in space hybrid MacCormack-based difference scheme (MacCormack 1969). Each simulation is performed up to a non-dimensional time of t ^⋆=200 which corresponds to two passes of the low-speed stream and requires 35,000 CPU h for the DNS and 250 CPU h for the LES on the IBM BladeCenter Linux Cluster. Both instantaneous and time-averaged data are presented to make qualitative and quantitative assessments of the temporal evolution and spatial structure of the particle field. The time-averaged data have been represented with an “overbar.”

The filter used in performing the LES is a top-hat or box-filter given by

where the grid filter width is Δ _H =15×Δ _DNS and the test filter width is The turbulent Schmidt numbers in the MKEV simulations are Sc_t =0.7, , and the SGS eddy-viscosity constant is set to C_k =0.005.

3.4. Fluid and Chemical Fields

Cross-stream profiles of the time-averaged velocity U/U_o are shown in Figure 2a. The velocity is plotted for both LES and DNS at x/D=5, x/D=10, and x/D=20. The figure shows little variation between the DNS and LES profiles at x/D=5. At x/D=10 jet diameters the DNS begins to widen, or spread in the cross-stream y-direction, and slow as the high-speed and low-speed streams mix. The u-velocity is greater in the LES and the width of the jet is smaller. At x/D=20, the DNS jet is over three times its original width and has slowed significantly as the eddies begin to pair and merge. The LES predicts a narrower jet width and flows that are 8% faster in the center than those found in DNS.

FIG. 2 Cross-stream profiles of time-averaged (a) streamwise velocity,

, and (b) conserved scalar,

, at three different downstream locations.

Figure 2b shows cross-stream profiles of the conserved scalar at x/D=5, x/d=10, and x/D=20. At x/D=5, the LES and DNS agree fairly well with the LES profile being a bit narrower than that of the DNS. At x/d=10, the LES profile is again narrower than that of the DNS, reflecting the lack of large-scale mixing and entrainment. The LES has a peak scalar concentration of in the center of the jet while the DNS has a peak value of In this region, the profiles show that the LES results exhibit less mixing jet, and yields concentrations that are between 20% and 30% higher than the DNS. At x/D=20 the peak concentration of the LES and DNS are similar though the DNS again exhibits greater transport in the cross-stream y-direction.

Instantaneous contours of the conserved scalar are shown in Figure 3. The contours reveal significant differences in the streamwise distance at which eddy formation occurs in each of the flows. In the DNS, shown in Figure 3a, the first eddies form after x/D=5. However in the LES, shown in Figure 3b, eddy formation is not evident until x/D=10. Toward the end of the domain, both the LES and DNS results contain large-scale eddies. However, in the LES there is a large pocket of fluid, near the centerline, which is relatively “unmixed” with the surrounding vapor (illustrated by a high φ value). While this is an instantaneous view, it is characteristic of the flow development in the LES and DNS.

FIG. 3 Contours of the instantaneous z-direction averaged conserved scalar, φ: (a) DNS; (b) LES.

Cross-stream profiles of the time-averaged TiO ₂ vapor mass-fraction are shown in Figure 4. The TiO ₂ initially zero and is formed as the reactants come into contact. The TiO ₂ is initially found at the interface of the two streams but by x/D=20, TiO ₂ is found across the span of the jet. At x/D=5, the over-predicts the DNS with a peak value of . This peak value is some 20% more than the DNS and is located near y/D=±0.5. At x/d=10, The LES still predicts greater amounts of TiO ₂, compared to the DNS, but the discrepancy in the shear layers (y/D=±0.5) has decreased to 10%. However, near the jet centerline (y/D=0), the DNS predicts more than twice as much TiO ₂ as the LES. The cause of this discrepancy is the fact that the jet in the LES persists farther into to the domain, compared to the DNS, and the large-scale mixing that brings the TiCl ₄ and H ₂ O is reduced. In the DNS, the vortical structures that form near x/D=5, as evidenced in Figure 3a, act to bring the reactants into contact and TiO ₂ is produced. Further downstream, at x/D=20, the shear layers in the LES have merged and TiO ₂ is present across the the entire jet width. Comparing the LES and DNS values shows significant differences. At x/D=20, the LES over-predicts the amount of TiO ₂ present near y/D=0, with values near , while the DNS predicts values near . These trends are inline with previous research that concludes that the effects of the SGS chemical reactions – neglected in this work – is to decrease the rate of chemical reaction (Garrick 1995; Colucci et al. 1998; Garrick et al. 1999). As a result, the chemical field as predicted by the LES contains more product (TiO ₂ and HCl) and less reactants (TiCl ₄ and H ₂ O)—than the chemical field predicted by the DNS.

FIG. 4 Cross-stream profiles of the time-averaged TiO ₂ vapor mass-fraction,

, at three different downstream locations.

3.5. Particle Field

The nodal method solves for the particulate field as a function of space, time, and size. In presenting our results we will focus on four of the twelve bins which correspond to particles of sizes 2nm, 4nm, 8nm, and 16nm in diameter. This is done to elucidate the dynamics of different size particles while minimizing the number of figures presented (Modem and Garrick 2003). As the flow develops, the reactants are converted to TiO ₂, particles nucleate, collide/coagulate, grow and therefore move from lower-numbered to higher-numbered bins. Additionally, particles are dispersed and transported from the particle-laden stream to the particle-free stream due to the effects of convection and diffusion (Modem et al. 2002; Moody and Collins 2003; Miller and Garrick 2004; Garrick et al. 2006; Wang and Garrick 2005). In this work, our focus is on the qualitative and quantitative differences between results obtained via LES and those obtained via DNS as they pertain to assessing the performance of the LES approach. Descriptions of the dynamics of nanoparticle growth in shear flows are available elsewhere (Modem et al. 2002; Miller and Garrick 2004; Settumba and Garrick 2007).

The 2nm, 8nm, 16nm, and 32nm particle concentrations are shown in Figure 5. (The concentrations are represented as the logarithm of the time-averaged concentration. i.e., log(/cm ³).) Cross-stream profiles of the time-averaged 2nm particle concentration, are shown in Figure 5a. The figure reveals that the peak concentration remains relatively constant as the jet travels downstream. This suggests that the rate at which particles entering “bin 7” (due to mixing, nucleation, condensation and coagulation) is balanced by the rate at which they leave (due to mixing, condensation, and coagulation) (Wang and Garrick 2005). At x/D=5, there is only a narrow band of particles that forms in the shear layers of the flow. As the flow reaches x/D=10 the bands of particles widen as the shear layer increases in size and larger concentrations can be found in the jet center. At x/D=20, the shear layers break down resulting in one large band of particles with the peak concentrations found at the jet center. At x/D=5, the results predicted by LES under-predicts the DNS by roughly 20%. At x/D=10, the LES under-predicts the DNS by 30% in the shear layers, y/D=±0.5, and 87% near the jet centerline, y/D=0. Further downstream, t x/D=20, the discrepancy has reversed and the LES predicts a 2nm particle concentration of , 58% more than the DNS. However, away from the center of the jet, the LES predicts fewer 2nm diameter particles. This trend is consistent with that observed with the TiO ₂ mass-fraction. Between x/D=10 and x/D=20 the jet becomes turbulent and large-scale mixing increases reactant conversion, nucleation and growth. At x/D=20, the LES over-predicts the number of 2nm particles.

FIG. 5 Cross-stream profiles of the time-averaged particle number concentrations at three different downstream locations (Case 1): (a) d_k =2nm; (b) d_k =8nm; (c) d_k =16nm; (d) d_k =32nm.

The same general trend is observed with the 8nm and 16nm diameter particles, shown in Figures 5b and c, respectively. The discrepancy between the LES and DNS at x/D=5 and x/D=10 persists, but at x/D=20, the peak concentrations agree fairly well. Turning to the 32nm diameter particles, shown in Figure 5d, the discrepancy between the LES and DNS persists with the LES predicting 50% fewer 32 nm particles than the DNS at x/D=5. At x/d=10, the LES under-predicts the DNS but at x/D=20, the discrepancy is decreased with the LES predicting more 32nm particles. These trends can be explained in part by the differences in eddy formation between LES and DNS. The DNS results showed eddy formation beginning to occur at x/D=5, where as there is no eddy formation at this location for LES. The eddies formed in DNS help mix the fluid and the reactants which results in more particle nucleation and coagulation, which would result in higher particle concentrations. Once the jet core collapses, the increased coagulation due to the absence of the SGS interactions in the LES – – causes the LES to particle field to “catch up” (Das and Garrick 2010).

3.6. Mean Size and Geometric Standard Deviation

The mean particle diameter, d_m , is given by

and represents the mass-mean diameter Cross-stream profiles of the time-averaged, mean diameter, for cases 1 and 2 are shown in Figure 6. The profiles show that the mean diameter increases as the flow travels downstream. The largest particles are found at the edges of the jet where eddies are formed, and also where the particles are moving relatively slowly. The same trend was shown by (Miller and Garrick 2004). Figure 6a shows that at x/D=5 the results predicted by the LES match closely with those predicted by the DNS. At a downstream location of x/d=10, the LES under-predicts the mean diameter by approximately 10%. At x/D=20, the results predicted by the LES and DNS match closely for the jet width predicted by LES. The DNS predicts a much wider jet width, and the LES fails to capture the mass-mean diameters toward the outside of the jet. Figure 6b shows that as the TiCl ₄ concentration is increased (from 0.5% to 1%), the same trend is observed. The only noticeable difference is the increase in mean particle diameter. The difference in peak particle size, roughly 20%, suggests that the growth-rate predicted by LES is less than that of the DNS. This means that in utilizing LES to predict the operation of a particle synthesis reactor — especially in the near-field region — one would have to correct the performance of the LES to account for the delayed transition to turbulence. Without such corrections, LES (and RANS) would predict smaller particles, if only because the flow-field transitions from laminar to turbulent further downstream.

FIG. 6 Cross-stream profiles of the time-averaged mean diameter,

, at three different downstream locations: (a) Case 1; (b) Case 2.

In the absence of large-scale spatial transport, i.e., temporal evolution only, regardless of the initial size-distribution, a self-preserving size distribution is eventually attained in the absence of particle formation and depletion (Friedlander 2000; Hinds 1982; Lee 1966). The width of the size-distribution is typically characterized by the geometric standard deviation (GSD) given by

where

The self-preserving size distribution obtained using a nodal method where the volume of particles in successive bins doubles, has a GSD of σ _g =1.5 (Modem et al. 2002; Garrick et al. 2006; Miller and Garrick 2004). Cross-stream profiles of the time-averaged GSD, at three different x/D locations are shown in Figure 7. The results for case 1, shown in Figure 7a show that at any given downstream location, the peak values are found near the interface of the jet and co-flowing streams. At locations x/D=5 and x/D=10 this is shear layers, visible as twin peaks in the profiles, where eddy formation and growth occur. At x/D=20, the GSD is relatively flat across the width of the jet. At the two earlier locations, the LES and DNS agree fairly well — both in terms of magnitude and distribution across the jet width. At x/D=20, however, the LES is once again under-predicting the extent of the jet that contains particles. Additionally, the DNS is predicts a comparatively high value of near y/D=5, whereas the LES predicts a value of The results for case 2 show similar trends, but with different GSD values. At x/D=5 and x/D=10, the LES and DNS agree fairly well but with the values slightly larger than in case 1. At x/D=20 the LES over predicts slightly in the region −3<y/D<2.5. Generally the discrepancy between the LES and DNS in case 2 is similar to the that observed in case 1.

FIG. 7 Cross-stream profiles of the time-averaged geometric standard deviation,

, at three different downstream locations: (a) Case 1; (b) Case 2.

Figure 8 shows instantaneous contours of mean diameter, d_m , and GSD, σ _g , plotted on an iso-surface of vorticity. The mean diameter values, shown in Figure 8a reveal small particles, uniformly distributed in the z-direction up to location x/D=15. Farther downstream, the eddies merge and form smaller scale structures – containing a wide range of mean diameters – which vary significantly across the z-direction. That is, the spanwise inhomogeneity of the particle size increases significantly after between x/D=15 and x/D=20. This behavior is reflected in the GSD as Figure 8b shows a similar trend. Once the flow breaks down into small-scale structures the GSD shows increases significantly and exhibits greater variation. Figure 8b shows peak GSD values of σ _g =6. This value is roughly four times that of the self-preserving distribution and it is significantly higher than those predicted by the time-averaged values, shown in Figure 7. The time-averaged profiles, predict GSD values less than This illustrates that time-averaging of the GSD suggests lower values than are actually manifest at any particular instant in time.

FIG. 8 Instantaneous contours of (a) mean diameter, d_m , and (b) geometric standard deviation, σ _g , on an iso-surface of vorticity magnitude, |ω|=1.8.

4. Summary and Conclusions

In this work, large eddy simulation (LES) and direct numerical simulation (DNS) of coagulating nanoparticles in incompressible, temporally developing mixing layers were performed. The filtered incompressible Navier-Stokes equations as well as the filtered nodal approximation to the general dynamic equation were solved for in a temporally and spatially accurate manner. Gradient-diffusion type closures were used to model the subgrid-scale (SGS) stress, the fluid-species fluxes and the fluid-particle flux. The effects of the unresolved SGS particle-particle coagulation terms were neglected.

The results show that the fluid field predicted by the LES initially exhibits less mixing and entrainment that the DNS. As a result the LES jet does not entrain as much of the background fluid as the DNS jet and therefore does not grow as much. This is a well known tendency with the Smagorinsky-type closures (Smagorinsky 1963; Germano et al. 1991; Garrick et al. 1999; Sagaut 2001). The reduced entrainment also means that the flow-through time, or the residence time within the computational domain is less in the LES as compared to the DNS. Turning to the particle field, the results show that the LES performs fairly well in predicting the particle number concentrations. The largest discrepancies are found in the region of the steep gradients. This again is due to the lack of entrainment, growth, and spreading of the LES jet (compared to the DNS). Upstream of the jet collapse, the LES predicts fewer particles of all sizes. However, once the core of the jet collapses and the flow transitions to a turbulent jet, the LES recovers and predicts concentrations similar to that of the DNS. The mean particle size predicted by the LES agrees fairly well with the DNS with the caveat that the LES jet is narrower than the DNS. So, across the entire jet, the DNS predicts larger particles than the LES. However, where the two flows do overlap, the LES matches the DNS quite well. This may suggest that if the large-scale mixing and entrainment were to be captured by the LES, the performance assessment of the LES would be more favorable. That is not certain because as entrainment and mixing is increased, more of the precursor is converted into nanoparticles and the growth-rate would increase. As a result the matter is not clear. Likewise for the geometric standard deviation (GSD), the LES and DNS values agree fairly well. The GSD values upstream of the jet collapse are qualitatively very similar. Further downstream, where the flow undergoes transition to turbulence, the differences begin to occur. As the precursor concentration increases, the differences become more significant, though they remain qualitatively similar. This trend is currently under investigation in a number of different flow configurations and an a priori analysis using DNS data also suggests this effect to be the true (Das and Garrick 2010).

Computationally, the LES produces results at a much reduced compute times. Each DNS required roughly 450 CPU-h on a SGI-3800 computer, whereas each LES required roughly 0.33 CPU-h. The performance differential is more than a factor of 1000. This suggests that LES may be a practical tool for simulating the formation and growth of nanoparticles in turbulent flows (Weier and Garrick 2005; Yu et al. 2008). What is lost in this type of simulation are small scale dynamics and features. Models for the SGS or turbulent fluid-particle, scalar-particle and particle-particle interactions are needed. A good example is the inclusion of finite-rate sintering, which affects the formation of hard agglomerates. Sintering rates are highly dependent on the temperature field and in turbulent reacting flows, these require accurate capture/resolution (Wegner and Pratsinis 2003; Park and Rogak 2003; Heine and Pratsinis 2007; Garrick and Wang 2010). Grid-free approaches such as the probability-based, filtered density function maybe suitable in both accounting for the unresolved terms as well as removing oscillations (Colucci et al. 1998; Garrick et al. 1999; Givi 2006; Garmory and Mastorakos 2008). However, for isothermal flows such as those considered in this study, given the degree of agreement between the LES and DNS, it is not clear that the superiority of the aforementioned approaches would facilitate the capture of the small-scale features. However, they should allow LES to provide more accurate predictions of integral quantities.

Acknowledgments

Funding for this work was provided by the University of Minnesota. Support for J. Loeffler was provided by the University of Minnesota's Undergraduate Research Opportunity Program. All computations were performed at the Minnesota Super-computing Institute.